I cant speak to the anchoring portion of your question but I can speak to the retrieving a mooring ball and pennants portion, which you did not ask about.
It really depends a lot on your local conditions; whether there is a lot of current or not, excessive wind, waves or chop.
For my area where there is typically a lot of current (Hudson River), the wind can be a big factor and trying to pick up a mooring (or anchoring) in a tight field of other boats it is important to me to not get out of control where I might hit one of the much more expensive boats then mine. I have found that having the main up can make our boat 'dance' about our mooring as it tries to sail from one side to the other but is actually tethered to the mooring. Unexpected jibes are dangerous in the best of situations and should be avoided but we have a lot of current that makes it much more difficult.
You can (and perhaps should) keep your main up when dropping
anchor in some situations if you are not going to turn on your engine beforehand but the safest way to handle a few tons of boat is to be able to do it both ways: (A) with no sails and engine only OR (B) with sails up and the engine in neutral (IN CASE) you need it. Plan (C): no engine with only sails is for people who have mastered plans (A) and (B), or when your engine quits unexpectedly.
The most important tools you have to choose which plan you want to employ are your senses and your own recognition of your skills to do so.
Choose wisely.